Locally connected Siegel Julia sets

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a polynomial $f$ of degree at least two, $f^n$ is its $n$-fold iterate, $K(f)=\{z\in\mathbb C:(f^n(z))_{n\geq0}\text{ is bounded}\}$ is its filled Julia set, and $J(f)=\partial K(f)$ is its Julia set. A space is locally connected if each point has a neighbourhood basis of connected sets. Let $P_\alpha(z)=z^2+e^{2\pi i\alpha}z$ for irrational $\alpha\in[0,1)$. It is a Siegel polynomial when a neighbourhood of $0$ is holomorphically conjugate to the irrational rotation $w\mapsto e^{2\pi i\alpha}w$; the maximal such rotation domain is its Siegel disk. Almost every angle refers to Lebesgue measure on $[0,1)$. The Hausdorff dimension $\dim_H E$ is the threshold exponent for Hausdorff measures $\mathcal H^s(E)$, defined using covers by sets of small diameter and sums of their $s$th powers. Planar Lebesgue measure is denoted by $m_2$.

Statement recorded in the supplied source.

Give an example of a Siegel polynomial $P_\alpha$ whose Julia set is provably locally connected. Is $J(P_\alpha)$ locally connected for Lebesgue-almost every choice of $\alpha$? What can be said about $\dim_HJ(P_\alpha)$? [S2]

Short English statement

Find a provably locally connected Siegel quadratic Julia set, decide whether this happens for almost every rotation angle, and determine what can be said about its Hausdorff dimension.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300034 (AMR-052-0034). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 John Milnor, "Problems on local connectivity", in Ben Bielefeld and Mikhail Lyubich (eds.), Problems in Holomorphic Dynamics (1992), Problem 9, pp. 22; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.